In 2-d cartesian coordinates, a vector with magnitude 20 forms a 30 degree angle above the positive x-axis. for reference, the negative y-axis represents a 3/2π radian angle from the positive x-axis in this coordinate system. by what absolute value percent would the x & y components of this vector change if the vector's magnitude was doubled and the vector was rotated another 110 degrees counter-clockwise?

Question

Mathematics

Squirt

5 July, 06:22

In 2-d cartesian coordinates, a vector with magnitude 20 forms a 30 degree angle above the positive x-axis. for reference, the negative y-axis represents a 3/2π radian angle from the positive x-axis in this coordinate system. by what absolute value percent would the x & y components of this vector change if the vector's magnitude was doubled and the vector was rotated another 110 degrees counter-clockwise?

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Ann Simpson · Answer 1

The Cartesian coordinates of the first vector are

20 (cos (30°), sin (30°))

After the vector is multiplied by 2∠110°, its Cartesian coordinates are

40 (cos (140°), sin (140°))

The x-component is larger than it was, but in the opposite direction. The percentage increase in the magnitude of the x-component is

x-change = (|40cos (140°) / (20cos (30°)) | - 1) * 100%

x-change ≈ 76.9%

The y-component is also larger than it was, but still in the + y direction. The percentage change in the magnitude of the y-component is

y-change = (|40sin (140°) / (20sin (30°)) | - 1) * 100%

y-change ≈ 157.1%